Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.09529

Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics

Juntao Wang·jieqiong Shi·Mei Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

The main aim of this paper is to introduce a Frobenius--Galois expansion of the substructural logic $\mathbf{FL}_{ew}$ and develop its algebraic and categorical semantics. The resulting logic, denoted by $\mathbf{FL}^{\mathbf{FGC}}_{ew}$, is obtained by adjoining a pair of unary connectives forming a Galois connection and satisfying suitable Frobenius-type compatibility conditions. Firstly, we prove that $\mathbf{FL}^{\mathbf{FGC}}_{ew}$ is algebraizable in the sense of Blok and Pigozzi and identify its equivalent algebraic semantics with the variety of Frobenius-adjoint residuated lattices, establishing the conservativity over $\mathbf{FL}_{ew}$ and relating its finite model property to residual finiteness of finitely generated free algebras. Secondly, for the distributive setting, we lift the Kalman construction to the Frobenius-adjoint algebraic framework. More precisely, we establish a categorical equivalence between Frobenius-adjoint residuated distributive lattices and Frobenius-adjoint $c$-differential residuated distributive lattices with the condition $\mathbf{CK}$. This equivalence yields a positive-cone representation of the former structures and, in turn, a logical counterpart of the categorical correspondence. Finally, for the distributive extension $\mathbf{FL}^{\mathbf{FGC},d}_{ew}$, we prove that derivability, validity over Frobenius-adjoint residuated distributive lattices, and validity over the corresponding positive cones determine the same consequence relation, and further show that this correspondence preserves equational and quasi-equational consequence, conservativity, and finite countermodels. These results provide a unified algebraic, categorical, and logical framework for Frobenius--Galois expansions of substructural logics.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Juntao Wang, jieqiong Shi, Mei Wang. 2026-09-08. Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics. https://arxiv.org/abs/2609.09529

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bluebirds and mockingbirds cannot produce a fixed-point combinator

Let $B$ be the bluebird combinator with reduction rule $Bxyz \to_{w} x\left(yz\right)$, let $M$ be the mockingbird combinator with reduction rule $Mx \to_{w} xx$, and let $I$ be the identity bird combinator with reduction rule $Ix \to_{w} x$. A fixed-point combinator, called a sage bird by Smullyan, is a closed term $Y$ such that, for a fresh variable $x$, $Yx$ is equivalent to $x\left(Yx\right)$ under these reduction rules. For a fixed variable $x$, we construct an invariant $\mathrm{Tr}_{x}\left(u\right)$ of a $BMI$-term $u$ with respect to $\to_{w}$. This invariant traces the occurrences of $x$ in the leftmost-innermost reduction sequence of $u$. We then prove that $\mathrm{Tr}_{x}\left(Yx\right) \neq \mathrm{Tr}_{x}\left(x^{r}\left( Yx \right)\right)$ for every $x$-free $BMI$-term $Y$ and every $r\geq 1$. Consequently, there exists no fixed-point combinator in $BMI$-combinatory logic. This provides a negative answer to the problem posed by Smullyan in 1985.

math.LO↗
arXiv2026

Pointwise provable equality and the failure of composition

Montagna (1989) and Di Paola--Montagna (1991) claim that the algebraic systems $S'$ and $S'_T$, respectively, are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two program indices that are pointwise provably equal in $T$ but yield inequivalent composites when each is run after the same program. Montagna's $S'$ is the case $T=\mathrm{PA}$. The failure already occurs for partial maps from $ω$ to itself. Weak totality and the proposed range assignment also depend on the choice of representatives. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $Π^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $Σ^0_1$-sound and the universal relation otherwise.

math.LO↗
arXiv2026

Compactness via Consistency Properties

We will use consistency properties to characterize strongly compact cardinals, first showing an adequate Model Existence Theorem for larger fragments.

math.LO↗